Quiver algebras: A quiver is a directed graph with the allowance of loops and multiple arrows. A quiver \(Q\)  is sometimes denoted as a quadruple \((Q_0,Q_1,o,t)\)  where \(Q_0\)  is the set of vertices in \(Q\) , \(Q_1\)  is the set of arrows in \(Q\) , and \(o,t: Q_1 \longrightarrow Q_0\)  are maps which assign to each arrow \(a\in Q_1\) , its origin vertex \(o(a)\)  and terminal vertex \(t(a)\)  in \(Q_0\) . A path in \(Q\)  is a sequence of arrows \(a = a_1 a_2 \cdots a_{n-1} a_n \)  such that the terminal vertex of \(a_{i}\)  is the same as the origin vertex of \(a_{i+1}\) , using the convention of concatenating paths from left to right. The quiver algebra or path algebra \(kQ\)  is defined as a vector space having all paths in \(Q\)  as a basis. Vertices are regarded as paths of length 0, an arrow is a path of length 1, and so on. We take multiplication on \(kQ\)  as concatenation of paths. Two paths \(a\)  and \(b\)  satisfy \(ab=0\)  if \(t(a)\ne o(b)\) . This multiplication defines an associative algebra over \(k\) . By taking \(kQ_i\)  to be the \(k\) -vector subspace of \(kQ\)  with paths of length \(i\)  as basis, \(kQ = \bigoplus _{i\ge 0} kQ_i\)  can be viewed as an \(\mathbb {N}\) -graded vector space. A relation on a quiver \(Q\)  is a linear combination of paths from \(Q\)  having the same origin vertex and terminal vertex. A quiver together with a set of relations is called a quiver with relations. Letting \(I\)  be an ideal of the path algebra \(kQ\) , we denote by \((Q,I)\)  the quiver \(Q\)  with relations \(I\) . The quotient \(\Lambda = kQ/I\)  is called the quiver algebra associated with \((Q,I)\) . Let \(\Lambda = \bigoplus _{i\ge 0}\Lambda _i\)  be a grading on \(\Lambda \) . If \(\Lambda \)  is Koszul and \(\Lambda _0\)  is isomorphic to \(k\)  or copies of \(k\) , \(\Lambda _0\)  has a linear graded projective resolution \(\mathbb {L}\)  as a right \(\Lambda \) -module [1]}, [2]}.
